\[ d_i = \sum^g_{j = 1} x_{ij} \]
igraph::degree(er_example) %>% summary()## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 3.0 8.0 11.0 10.6 12.0 18.0
igraph::degree(ws_example) %>% summary()## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 7 9 10 10 11 13
igraph::degree(b_example) %>% summary()## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 11.00 11.00 12.00 20.68 17.00 90.00
\[ d'_i = \frac{d_i}{g - 1} \]
\[ D = \frac{\sum_{i=1}^g [d_{max} - d_i]}{(g - 2)(g - 1)} \]
igraph::centr_degree(er_example)$centralization## [1] 0.07474747
igraph::centr_degree(ws_example)$centralization## [1] 0.03030303
igraph::centr_degree(b_example)$centralization## [1] 0.3536374
\[ c_i = [\sum_{j=1}^g d_{ij}]^{-1} \]
\[ c'_i = (g - 1)c_i \]
#igraph::closeness(er_example, normalized = TRUE) %>% summary()
igraph::closeness(ws_example, normalized = TRUE) %>% summary()## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.3333 0.3623 0.3808 0.3826 0.3992 0.4605
igraph::closeness(b_example, normalized = TRUE, mode = c("in")) %>% summary()## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.01000 0.01000 0.01020 0.03891 0.01285 0.91670
igraph::closeness(b_example, normalized = TRUE, mode = c("all")) %>% summary()## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.5294 0.5294 0.5323 0.5668 0.5470 0.9167
\[ C = \frac{\sum_{i=1}^g c'_{max} - c'_i}{(g - 2)(g - 1)/(2g - 3)} \]
igraph::centr_clo(er_example, normalized = TRUE)$centralization ## [1] 0.1322996
igraph::centr_clo(ws_example, normalized = TRUE)$centralization ## [1] 0.1581056
igraph::centr_clo(b_example, normalized = TRUE, mode = c("all"))$centralization ## [1] 0.7104668
\[ B (n_i) = \sum_{i \neq j \neq k} \frac{g_{jk}(n_i)}{g_{jk}} \]
where:
\[ B' (n_i) = \frac{B (n_i) }{[(g - 1)(g - 2)/2]} \]
igraph::betweenness(er_example, normalized = TRUE, directed = FALSE) %>%
summary()## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0006065 0.0066530 0.0106600 0.0121400 0.0148100 0.0381700
igraph::betweenness(ws_example, normalized = TRUE, directed = FALSE) %>%
summary()## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.001963 0.005955 0.012500 0.016590 0.023800 0.065370
igraph::betweenness(b_example, normalized = TRUE, directed = TRUE) %>%
summary()## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0000000 0.0000000 0.0006321 0.0012750 0.0018810 0.0072570
\[ B = \frac{2\sum_{i=1}^g[b_{max} - b_i]}{[(g-1)^2(g-2)]} \]
\[ e_i = \frac{1}{\lambda}\sum_{j: j \neq i} x_{ij}e_j \]
where:
igraph::evcent(er_example, directed = FALSE)$vector %>% summary()## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.1409 0.4003 0.5189 0.5281 0.6252 1.0000
igraph::evcent(b_example, directed = TRUE)$vector %>% summary()## Warning in .Call("R_igraph_eigenvector_centrality", graph, directed,
## scale, : At centrality.c:344 :graph is directed and acyclic; eigenvector
## centralities will be zeros
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0 0 0 0 0 0
igraph::evcent(b_example, directed = FALSE)$vector %>% summary()## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.1692 0.2430 0.2681 0.3406 0.3107 1.0000
\[ E = \frac{\sum_{i = 1}^g(e_{max}-e_i)}{(1-e_i)} \]
igraph::centr_eigen(er_example)$centralization ## [1] 0.4815106
igraph::centr_eigen(ws_example)$centralization ## [1] 0.309248
igraph::centr_eigen(b_example, directed = FALSE)$centralization ## [1] 0.6728771